Answer
$4cm, 2cm$
Work Step by Step
1. Let the radii of the two circles be $x$ and $y$, based on the given conditions, we can get:
$\begin{cases}2\pi(x+y)=12\pi \\ \pi( x^2+y^2)=20\pi \end{cases}$ or $\begin{cases}x+y=6 \\ x^2+y^2=10\end{cases}$
2. The 1st equation gives $y=6-x $, use it in the 2nd equation to get $x^2+36-12x+x^2=20$ or $x^2-6x+8=0$ or $(x-4)(x-2)=0$ thus $x=2,4$, finally, use the 1st equation to get $y=4,2$
3. We have the solution(s) $4cm, 2cm$
